Orientations for DT invariants on quasi-projective Calabi-Yau $4$-folds

Abstract: Donaldson-Thomas type invariants in complex dimension $4$ have attracted a lot of attention in the past few years. I will give a brief overview of how one can count coherent sheaves on Calabi-Yau $4$-folds. Inherent to the definition of DT4 invariants is the notion of orientations on moduli spaces of sheaves/ perfect complexes. For virtual fundamental classes and virtual structure sheaves to be well-defined, one needs to prove orientability. The result of Cao-Gross-Joyce does this for projective CY $4$-folds. However, computations are more feasible in the non-compact setting using localization formulae, where the fixed point loci inherit orientations from global ones, and orientations of the virtual normal bundles come into play. I will explain how to use real determinant line bundles of Dirac operators on the double of the original Calabi-Yau manifold to construct orientations on the moduli stack of compactly supported perfect complexes, moduli schemes of stable pairs and Hilbert schemes. These are controlled by choices of orientations in K-theory and satisfy compatibility under direct sums. If time allows, I will discuss the connection between the sings obtained from comparing orientations and universal wall-crossing formulae of Joyce using vertex algebras.

algebraic geometrycombinatorics

Audience: researchers in the topic

( slides | video )

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Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html

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